vector subspace

Definition
Let V be a vector space over a field F,
and let W be a subset of V.
If W is itself a vector space,
then W is said to be a vector subspace of V.
If in addtition V≠W, then W is a proper vector subspace of V.

If W is a nonempty subset of V,
then a necessary and sufficient condition for W to be a subspace
is that a+γ⁢b∈W
for all a,b∈W and all γ∈F.

0.0.1 Examples

1.

Every vector space is a vector subspace of itself.

2.

In every vector space, {0} is a vector subspace.

3.

If S and T are vector subspaces of a vector space V,
then the vector sum